Response Matrix Properties and Convergence Implications for an Interface-Current Nodal Formulation

An analytic study was performed of the properties and the associated convergence implications of the response matrix equations derived via the widely used nodal expansion method. By using the DIF3D nodal formulation in hexagonal-z geometry as a concrete example, an analytic expression for the response matrix is first derived by using the hexagonal prism symmetry transformations. The spectral radius of the local response matrix is shown to be always <1. The l₂-norm of the response matrix is shown to be <1 in two-dimensional problems but not always <1 in three-dimensional problems. The elements of the response matrix are shown to not always be positive, and the l_∞-norm is not always <1. The spectral radius and the l₂- and l_∞-norms of the response matrix are found to increase as the removal cross section decreases. On the other hand, for a given removal cross section, each of these matrix norms takes its minimum at a certain diffusion coefficient and increases as the diffusion coefficient deviates from this value. Based on these matrix norms, sufficient conditions for the convergence of the iteration schemes for solving the response matrix equations are discussed. The range of node-height-to-hexagon-pitch ratios that guarantees a positive solution is derived as a function of the diffusion coefficient and the removal cross section.